Noise reducing/resolution enhancing signal processing method and system

ABSTRACT

A noise reduction/resolution enhancement signal processing method and system is disclosed, wherein the influence of noise spikes and gaps is substantially reduced. The data for the noise reduction may be amplitudes (b x ) measured at corresponding values (x) over a given domain D, wherein the data defines a composite wave form. The composite wave form is decomposed into instances of a discrete wave form, each having reduced noise amplitudes. A candidate point c in D for, e.g., an amplitude extreme is determined for each discrete wave form instance (having unknown amplitude). A minimization technique determines a first set of discrete wave form instances (having known amplitudes) by collapsing on the amplitudes (b x ) from above. A maximization technique determines a second set of discrete wave form instances (having known amplitudes) by rising up to the amplitudes (b x ) from below. For each point c, a determination is made as which of the corresponding instances from the first and second sets results in a better reduction in noise, and this determination is used to provide a resulting discrete wave form instance at c. Embodiments of the invention are disclosed for mass and light spectrometry, digital imaging and audio applications.

RELATED APPLICATIONS

The present application claims the benefit of the U.S. Provision PatentApplication Ser. No. 60/169,178 filed Dec. 6, 1999.

RELATED FIELD OF THE INVENTION

The present invention is related to a signal processing method andsystem for reducing noise and enhancing resolution of signal dataobtained from; e.g., spectrometer, a digital camera, or a digital soundrecorder.

BACKGROUND

Mass Spectrometry is an analytical technique used to identify compoundsbased on their molecular weights. There are several basic types of massspectrometers, all of which rely on similar basic principles. In massspectrometry, samples are ionized at a source and are selectivelyaccelerated using strong electromagnetic fields to a detector. Theresult is a discrete frequency distribution of mass to charge ratiosdetected by the mass spectrometer as a relative (i.e., detectordependent) detector impact intensity. The output is a spectrum ofintensity peaks as shown in FIG. 1.

One of the problems that often arise in mass spectrometry (and manyother signal processing applications) is the difficulty indistinguishing a real signal from noise produced when obtaining suchintensities. Furthermore, any given mass spectrometer has a limitedsampling resolution thus causing a single ionized mass to be displayedas a Gaussian distribution rather than a single discrete point. Thelimited resolution coupled with the noise problem often makes itdifficult to identify and separate non-noise signals that are weakand/or so close in proximity to other non-noise signals such that it isdifficult to accurately quantify the intensity of peaks that can be usedfor identifying the compounds. In particular, quantification of weak,noisy and close peaks is very difficult. More generally, low resolutionsignal data often occurs as a result of hardware limitations.Accordingly, an algorithm that would effectively increase resolutionwould be very desirable as it is far less costly than improving thehardware.

Accordingly, it would be advantageous to have a signal processingtechnique that could enhance the discrimination of such non-noise peaks.

SUMMARY

The present invention is a method and system for performing a signalprocessing technique that enhances the discrimination of non-noise waveforms in measurements of a plurality of inputs obtained from a process.In particular, the present invention decomposes or deconvolves a waveform (denoted herein as a composite wave form) obtained from theplurality of inputs into predetermined non-composite wave forms (denotedherein as discrete wave forms, or peaks). More particularly, the presentinvention can be used to decompose such composite wave forms in signalprocessing technologies such as mass spectrometry, or any othercomposite wave form outputting process/application wherein:

(a) the inputs are discrete over a known domain;

(b) the peaks may only occur at predetermined discrete value over thedomain;

(c) for each peak of the corresponding measurements for the inputs,there is an expected wave form for which the peak is a maximal value;

(d) the resolution is known by which measurements or readings for inputsto the wave form outputting process/application can be distinguished,i.e., resolution is the finest or smallest discrimination that isavailable for in measurements or readings from the wave form outputtingprocess/application.

In one embodiment, the present invention can be used for processing massspectra using linear programming, wherein:

(i) the expected discrete wave forms are Gaussian,

(ii) the predetermined resolution is that of the mass spectrometer beingused to obtain intensities for inputs of mass to charge ratios; i.e.,the predetermined resolution is the smallest difference in the m/z ratio(more generally, the input values used by the wave form outputtingprocess/application) that the mass spectrometer (more generally, thewave form outputting process/application) can distinguish differences inreadings of intensity/amplitude., and

(iii) the resulting peaks can only occur at the discrete points alongthe mass to charge ratio domain corresponding to atomic mass units.Accordingly, a result from the present invention is a deconvolvedspectrum of charge ratios with accurate relative abundances of compoundsas one skilled in the art will understand.

When an embodiment of the present invention utilizes linear programmingto “fit” a sum of Gaussian (discrete wave form) functions to a given(composite wave form) spectrum where each of the discrete wave formfunctions is approximately centered on some multiple 1 through k of apredetermined value V, then the sum of the composite wave form is givenbelow: $\begin{matrix}{{f(x)} = {\sum\limits_{n = 2}^{k}{c_{n}^{\frac{- {({x - n})}^{2}}{a^{2}}}}}} & (1.1)\end{matrix}$

where x is an independent parameter of mass to charge ratios, k is thelargest possible center for a discrete wave in the region, c_(n) is theheight (or equivalently, intensity or amplitude) of the n^(th) discretewave form, and a is proportional to the discrete wave form width, andwhere each discrete wave form is of the form: $\begin{matrix}{{g(x)} = ^{\frac{- x^{2}}{a^{2}}}} & (1.2)\end{matrix}$

where a is proportional to the width of the distribution.

In particular, for a given composite wave form to be decomposed, thepresent invention performs the following steps:

a) Minimizes the vector {overscore (c)}=[c₁, c₂, c₃, . . . , c_(k)]subject to each of the f(x_(i))≧b_(i), where b_(i) is the intensity orheight of the given spectrum at x_(i).

b) Maximizes the vector {overscore (c)} subject to each of thef(x_(i))≦b_(i), where b_(i) is the intensity or height of the givenspectrum at x_(i).

Note that for mass spectrometry, atomic masses are generally quite closeto integral values of one another (i.e., 1 for hydrogen, 2 for helium,etc.). Moreover, it is generally possible to determine the largestionization charge that can be obtained for a mass spectrometer accordingto the ionization energy that the mass spectrometer applies to asubstance being assayed. In particular, ion charges are typicallybetween +1 and +10. Accordingly, if ions having charges between +1 and+10 are potentially generated, then the predetermined value V mentionedabove may be, e.g., a mass-to-charge ratio such as 1/(2*3*5*7)=1/210 ofan atomic mass. Note that to simplify the description herein (andwithout loss of generally), it can be assumed that that thepredetermined value V is one, and accordingly the centers of thediscrete wave forms are integer values.

To provide additional background material for understanding the presentinvention, the following references are fully incorporated herein byreference:

1. http://www.asms.org/—The American Society for Mass Spectrometryhomepage;

2. Nash, Stephen G., Sofer Ariela, Linear and Nonlinear Programming.McGraw-Hill 1996;

3. Stephen Wolfram The Mathematica® Book Fourth Edition Wolfram Research1998; and

4. Applications Guide, Micromass, 1998.

Other features and benefits of the present invention will become evidentfrom the accompanying figures and Detailed Description.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a graph of a discrete frequency distribution of mass tocharge ratios detected by the mass spectrometer as a relative (i.e.,detector dependent) detector impact intensity. The output shows aspectrum of intensity peaks.

FIG. 2 shows a graph of the minimization operation (2.1) (graph 204) andmaximization operation (2.2) (graph 216) together with a representationof a composite wave form having both a noise spike 208 and a gap 212.

FIG. 3 illustrates a decomposition of a composite wave form intodiscrete wave forms 304, 308 and 312, wherein there is cross-talkbetween the discrete wave forms.

FIG. 4 is a graph of an idealized test set of a composite wave form withno random noise and a very high resolution (e.g., 0.5) upon which theminimization operation (2.1) and the maximization operation (2.2) areapplied as an illustrative example of the signal processing stepsperformed by a linear programming embodiment of the present invention.

FIG. 5 is a graph of a low resolution test set (TEST₁) of a compositewave form. where intrusion of one discrete wave form instance on thenext is considerable. In particular, the width at half maximum intensityknown in the art as Full-Width-Half-Max (FWHM) is set to 1.5 units forthis test set. This translates into approximately 30% of the intensityof one peak still being present in an adjacent peak's maximal value.

FIG. 6 is a graph of a low resolution test set (TEST₂) of a compositewave form. where one peak is much stronger than an adjacent peak. Oftenwhen this occurs, the smaller peak is masked in the composite wave formso that it is not detectable. In addition, the contribution of thelarger peak may extend noticeably to non-adjacent weak peaks. Tosimulate such a condition, the resolution of the TEST₂ test set is setto 0.7 FWHM units (wherein there was only 0.35% intrusion of one peak onthe next), and the intensity of the middle peak is 100. Note that ascaled down version of the graph is also provided in this figure so thatthe entire graph can be represented.

FIG. 7 is a graph of a test set (TEST₃) of a composite wave form. wherethis test set includes a spike of 2 intensity units added to b₈. Theresolution is set to 0.5 FWHM units (only 0.0015% intrusion of one peakto the next).

FIG. 8 is a graph of a test set (TEST₄) of a composite wave form. wherethis test set includes a decrease of 2 intensity units (a gap) at b₇(the middle of the third peak). The resolution for this data set is setto 0.5 FWHM units (only 0.0015% intrusion of one peak to the next).

FIG. 9 is a graph of a test set (TEST₅) of a composite wave form. wherethis test set includes both poor resolution and a positive noise spike(which in mass spectrometry is more likely than a negative spike).Accordingly, this test set tests the performance of the linearprogramming embodiment of the present invention under conditions thatmore closely mimic reality. The resolution of this test set is 1.5 FWHM(which implies a 30% intrusion of one peak on the next) and a positivenoise spike of 2 has been added to b₈.

FIG. 10 shows various graphs of data generated by the present inventionwhen it is assumed that there is limited cross-talk between the discretewave forms into which the composite wave form (represented by test setTEST₁) is decomposed. More particularly, the following graphs areillustrated in this figure: (a) a graph of the function f(x) defined in(1.1) as determined by the minimization operation (2.1) applied to testset TEST, (this graph is labeled “MIN”), (b) graphs of the correspondingdiscrete wave forms obtained from the minimization operation (2.1) (eachof these graphs is labeled “P_(MIN)”); (c) a graph of the function f(x)defined in (1.1) as determined by the maximization operation (2.2)applied to test set TEST₁ (this graph is labeled “MAX”), and (d) graphsof the corresponding discrete wave forms obtained from the maximizationoperation (2.2) (each of these graphs is labeled “P_(MAX)”).

FIG. 11 shows various graphs of data generated by the present inventionwhen it is assumed that there is continuous cross-talk between thediscrete wave forms into which the composite wave form (represented bytest set TEST₁) is decomposed. More particularly, the following graphsare illustrated in this figure: (a) a graph of the function f(x) definedin (1.1) as determined by the minimization operation (2.1) applied totest set TEST₁ (this graph is labeled “MIN”), (b) graphs of thecorresponding discrete wave forms obtained from the minimizationoperation (2.1) (each of these graphs is labeled “P_(MIN)”); (c) a graphof the function f(x) defined in (1.1) as determined by the maximizationoperation (2.2) applied to test set TEST₁ (this graph is labeled “MAX”),and (d) graphs of the corresponding discrete wave forms obtained fromthe maximization operation (2.2) (each of these graphs is labeled“P_(MAX)”).

FIG. 12 shows various graphs of data generated by the present inventionwhen it is assumed that there is limited cross-talk between the discretewave forms into which the composite wave form (represented by test setTEST₂) is decomposed. More particularly, the following graphs areillustrated in this figure: (a) a graph of the function f(x) defined in(1.1) as determined by the minimization operation (2.1) applied to testset TEST₂ (this graph is labeled “MIN”), (b) graphs of the correspondingdiscrete wave forms obtained from the minimization operation (2.1) (eachof these graphs is labeled “P_(MIN)”); (c) a graph of the function f(x)defined in (1.1) as determined by the maximization operation (2.2)applied to test set TEST₂ (this graph is labeled “MAX”), and (d) graphsof the corresponding discrete wave forms obtained from the maximizationoperation (2.2) (each of these graphs is labeled “P_(MAX)”). Note that ascaled down version of these graphs is also provided in this figure sothat the entire graph of each of (a) and (c) can be represented.

FIG. 13 shows various graphs of data generated by the present inventionwhen it is assumed that there is continuous cross-talk between thediscrete wave forms into which the composite wave form (represented bytest set TEST₂) is decomposed. More particularly, the following graphsare illustrated in this figure: (a) a graph of the function f(x) definedin (1.1) as determined by the minimization operation (2.1) applied totest set TEST₂ (this graph is labeled “MIN”), (b) graphs of thecorresponding discrete wave forms obtained from the minimizationoperation (2.1) (each of these graphs is labeled “P_(MIN)”); (c) a graphof the function f(x) defined in (1.1) as determined by the maximizationoperation (2.2) applied to test set TEST₂ (this graph is labeled “MAX”),and (d) graphs of the corresponding discrete wave forms obtained fromthe maximization operation (2.2) (each of these graphs is labeled“P_(MAX)”). Note that a scaled down version of these graphs is alsoprovided in this figure so that the entire graph of each of (a) and (c)can be represented.

FIG. 14 shows various graphs of data generated by the present inventionwhen it is assumed that there is limited cross-talk between the discretewave forms into which the composite wave form (represented by test setTEST₃) is decomposed. More particularly, the following graphs areillustrated in this figure: (a) a graph of the function f(x) defined in(1.1) as determined by the minimization G operation (2.1) applied totest set TEST₃ (this graph is labeled “MIN”), (b) graphs of thecorresponding discrete wave forms obtained from the minimizationoperation (2.1) (each of these graphs is labeled “P_(MIN)”); (c) a graphof the function f(x) defined in (1.1) as determined by the maximizationoperation (2.2) applied to test set TEST₃ (this graph is labeled “MAX”),and (d) graphs of the corresponding discrete wave forms obtained fromthe maximization operation (2.2) (each of these graphs is labeled“P_(MAX)”).

FIG. 15 shows various graphs of data generated by the present inventionwhen it is assumed that there is continuous cross-talk between thediscrete wave forms into which the composite wave form (represented bytest set TEST₃) is decomposed. More particularly, the following graphsare illustrated in this figure: (a) a graph of the function f(x) definedin (1.1) as determined by the minimization operation (2.1) applied totest set TEST₃ (this graph is labeled “MIN”), (b) graphs of thecorresponding discrete wave forms obtained from the minimizationoperation (2.1) (each of these graphs is labeled “P_(MIN)”); (c) a graphof the function f(x) defined in (1.1) as determined by the maximizationoperation (2.2) applied to test set TEST₃ (this graph is labeled “MAX”),and (d) graphs of the corresponding discrete wave forms obtained fromthe maximization operation (2.2) (each of these graphs is labeled“P_(MAX)”).

FIG. 16 shows various graphs of data generated by the present inventionwhen it is assumed that there is limited cross-talk between the discretewave forms into which the composite wave form (represented by test setTEST₄) is decomposed. More particularly, the following graphs areillustrated in this figure: (a) a graph of the function f(x) defined in(1.1) as determined by the minimization operation (2.1) applied to testset TEST₄ (this graph is labeled “MIN”), (b) graphs of the correspondingdiscrete wave forms obtained from the minimization operation (2.1) (eachof these graphs is labeled “P_(MIN)”); (c) a graph of the function f(x)defined in (1.1) as determined by the maximization operation (2.2)applied to test set TEST₄ (this graph is labeled “MAX”), and (d) graphsof the corresponding discrete wave forms obtained from the maximizationoperation (2.2) (each of these graphs is labeled “P_(MAX)”).

FIG. 17 shows various graphs of data generated by the present inventionwhen it is assumed that there is continuous cross-talk between thediscrete wave forms into which the composite wave form (represented bytest set TEST₄) is decomposed. More particularly, the following graphsare illustrated in this figure: (a) a graph of the function f(x) definedin (1.1) as determined by the minimization operation (2.1) applied totest set TEST₄ (this graph is labeled “MIN”), (b) graphs of thecorresponding discrete wave forms obtained from the minimizationoperation (2.1) (each of these graphs is labeled “P_(MIN)”); (c) a graphof the function f(x) defined in (1.1) as determined by the maximizationoperation (2.2) applied to test set TEST₄ (this graph is labeled “MAX”),and (d) graphs of the corresponding discrete wave forms obtained fromthe maximization operation (2.2) (each of these graphs is labeled“P_(MAX)”).

FIG. 18 shows various graphs of data generated by the present inventionwhen it is assumed that there is limited cross-talk between the discretewave forms into which the composite wave form (represented by test setTEST₅) is decomposed. More particularly, the following graphs areillustrated in this figure: (a) a graph of the function f(x) defined in(1.1) as determined by the minimization operation (2.1) applied to testset TEST₅ (this graph is labeled “MIN”), (b) graphs of the correspondingdiscrete wave forms obtained from the minimization operation (2.1) (eachof these graphs is labeled “P_(MIN)”); (c) a graph of the function f(x)defined in (1.1) as determined by the maximization operation (2.2)applied to test set TEST₅ (this graph is labeled “MAX”), and (d) graphsof the corresponding discrete wave forms obtained from the maximizationoperation (2.2) (each of these graphs is labeled “P_(MAX)”).

FIG. 19 shows various graphs of data generated by the present inventionwhen it is assumed that there is continuous cross-talk between thediscrete wave forms into which the composite wave form (represented bytest set TEST₅) is decomposed. More particularly, the following graphsare illustrated in this figure: (a) a graph of the function f(x) definedin (1.1) as determined by the minimization operation (2.1) applied totest set TEST₅ (this graph is labeled “MIN”), (b) graphs of thecorresponding discrete wave forms obtained from the minimizationoperation (2.1) (each of these graphs is labeled “P_(MIN)”); (c) a graphof the function f(x) defined in (1.1) as determined by the maximizationoperation (2.2) applied to test set TEST₅ (this graph is labeled “MAX”),and (d) graphs of the corresponding discrete wave forms obtained fromthe maximization operation (2.2) (each of these graphs is labeled“P_(MAX)”).

FIGS. 20A through 20C is a flowchart showing the high level stepsperformed by the present invention when determining the amplitudes andcenters of a plurality of discrete wave forms into which a compositewave form is decomposed.

FIG. 21 is a block diagram showing the components of a mass spectrometrysystem 2100 which is an embodiment of the present invention, wherein thesystem 2100 reduces the noise and/or enhances the resolution of massspectrometry data obtained from a sample compound or material.

FIG. 22 is a block diagram showing the components of a lightspectrometry system 2200 which is an embodiment of the presentinvention, wherein the system 2200 reduces the noise and/or enhances theresolution of light spectrometry data obtained from lightemitted/reflected from a sample compound or material.

FIG. 23 is a block diagram showing the components of an imageenhancement system 2300 which is an embodiment of the present invention,wherein the system 2300 reduces the noise and/or enhances the resolutionof image data obtained from a digital camera 2304.

FIG. 24 is a block diagram showing the components of an audioenhancement system 2400 which is an embodiment of the present invention,wherein the system 2400 reduces the noise and/or enhances the resolutionof audio data obtained from a digital sound recorder 2404.

DETAILED DESCRIPTION

A first embodiment of the present invention for performing massspectrometry data analysis will be described. Accordingly, the input tothis embodiment is a spectrum from either a simulation of a massspectrum, or from a mass spectrometer. The input may be in the form of alist of pairs, with each pair containing an intensity value and itscorresponding mass to charge ratio. Substantially all mass spectrometerssample at a rate much higher than integer masses. Thus, the input willhave many samples for each mass unit typically at equal intervals oversome predetermined range such as 20 to 200. This input can be written asa matrix in the following form: $\begin{bmatrix}{{Intensity}_{1}\quad \ldots \quad {Intensity}_{n}} \\{{M/Z_{1}^{+}}\quad \ldots \quad {M/Z_{n}^{+}}}\end{bmatrix}$

The first row of this matrix will be referred to as the {overscore(b)}=[b₁, b₂, . . . , b_(n)] vector.

At a high level, the first embodiment can be summarized as a method andsystem for performing at least one of the following operations fordetermining the peak amplitudes c_(i): $\begin{matrix}{{{{Minimize}\quad {\sigma \left( \overset{\_}{c} \right)}} = {\sum\limits_{i = 2}^{k}c_{i}}}{{Subject}\quad {to}\quad {the}\quad {{constraint}:{{f\left( \overset{\_}{x} \right)} \geq \overset{\_}{b}}}}{{and}/{or}}} & (2.1) \\{{{{Maximize}\quad {\sigma \left( \overset{\_}{c} \right)}} = {\sum\limits_{i = 2}^{k}c_{i}}}{{Subject}\quad {to}\quad {the}\quad {{constraint}:{{f\left( \overset{\_}{x} \right)} \leq \overset{\_}{b}}}}} & (2.2)\end{matrix}$

where f(x) is given in (1.1) above in the Summary Section. Moregenerally, the minimization and maximization operations may be performedon substantially any functional σ (i.e., that maps vectors to a realand/or complex number) that monotonically increases with increasingvalues of each c_(i) in the range of likely intensity/amplitude values(e.g., at the range including the b_(i)'s), and correspondingly alsomonotonically decreases with decreasing values of each c_(i) in therange of likely intensity/amplitude values (e.g., at the range includingthe b_(i)'s). Accordingly, note that the function f(x) as defined in(1.1) may be used for σ. Moreover, note that many approximations to f(x)as defined in (1.1) may be used for σ. In particular, linearapproximations to f(x) as defined in (1.1), wherein such linearapproximations can be used in linear programming can also be used for σ.In fact,${\sigma \left( \overset{\_}{c} \right)} = {\sum\limits_{i = 2}^{k}{w_{i}*c_{i}}}$

where w_(i) are positive is linear approximation of f(x) as defined in(1.1), as one skilled in the art will understand, and minimizing(maximizing)${\sigma \left( \overset{\_}{c} \right)} = {\sum\limits_{i = 2}^{k}c_{i}}$

yields the same result as minimizing (maximizing)${\sigma \left( \overset{\_}{c} \right)} = {\sum\limits_{i = 2}^{k}{w_{i}*{c_{i}.}}}$

Note that (2.1) and (2.2) are quite different, each with its ownadvantage. The minimization problem produces an upper bound on thespectrum intensities, while the maximization produces a lower bound onthe spectrum intensities.

Regarding noise in mass spectrometry data input to the first embodiment,there may be a fundamental distinctions in the results from each of theoperations (2.1) and (2.2). However prior to describing suchdistinctions, it is important to identify several different types ofnoise, and how such noise comes about.

One type of noise, is a called a noise spike. A noise spike is apositive, sharp rise in the spectral signal intensity caused by ananomaly, which causes a substantially falsely high reading at onemeasurement (e.g., if a measurement x_(i) has a falsely high b_(i), thenit is presumed that within a neighborhood {x such that(x_(i)−δ)<=x<=(x_(i)−δ) for δ>the sampling frequency} each b_(j)corresponding to such an x (other than x_(i)) in the neighborhood issubstantially a correct intensity reading and not falsely high). Forexample, a spike may be defined to be a reading that is at least 1.5times greater than what the corresponding correct reading would be.Spikes may be produced in a mass spectrometer due to, e.g., smallelectrical surges, black body radiation, poor vacuum, etc.

A second, more rare type of noise is called a noise gap. A noise gap isa negative, sharp decline in the spectral signal intensity caused by ananomaly. More specifically, a noise gap (also denoted simply gap herein)is a falsely low reading at one measurement (e.g., if a measurementx_(i) has a falsely low b_(i), then it is presumed that within aneighborhood {x such that (x_(i)−δ)<=x<=(x_(i)−δ) for δ>the samplingfrequency} each b_(j) corresponding to such an x (other than x_(i)) inthe neighborhood is substantially a correct intensity reading and notfalsely low). For example, a gap may be defined to be a reading that isat least 1.5 times lower than what the corresponding correct readingwould be. Generally in mass spectrometry this type of noise occurs lessoften since most generated spectral data are in fact many spectraaveraged together to reduce the signal to noise ratio. Since averagingis an additive process any noise gap in one spectrum is usually filledby averaging with others (all spectra have only non-negativeintensities). This is not true of noise spikes; in fact, in the extremecase, noise spikes may accumulate with averaging.

Moreover, it is important to note that spike and gap types of noise arenot limited to spectral signal data. Such noise types may occur invarious signal processing applications such as mass or lightspectrometry for: (a) absorption/transmission analysis for measuring aconcentration, and/or composition and/or atomic configuration ofsubstances or mixtures, (b) fluorescence, (c) Raman, (d) naturalmagnetic resonance (NMR). Additionally, the signal processing method ofthe present invention may also be used for enhancing a digital image orany image where the image is composed of discrete samples and theunderlying distribution is known.

As shown in the graph 204 (FIG. 2) of f(x), the corresponding to thec_(i)'s determined by the minimization operation of (2.1) may result inf(x) not closely fitting the intensities in a neighborhood of a noisespike (e.g., spike 208), wherein the term “closely fitting theintensities in a neighborhood” is understood to mean, e.g., that foreach b_(j) in the neighborhood, the absolute value of the deviationbetween f(x_(j)) and b_(j) is less than a predetermined percentage(e.g., 5%) of the intensity of b_(j). In particular, the minimizationoperation constrains f(x_(i)) to be at least as great as b_(i) andaccordingly such a spiked measurement 208 will not allow f(x) to closelyfit to adjacent intensity measurements. Thus, a falsely high result forthe amplitudes c_(i) is obtained. However, the minimization operation(2.1) can be ideal for noise gap problems (e.g., gap 212, FIG. 2), sinceadjacent measurements restrict the minimization from going into thegapped region (assuming, e.g., an appropriate condition such as the gapwas narrower than the resolution.

In contrast to the minimization operation (2.1), the f(x) and thecorresponding to the c_(i)'s determined by the maximization operation(2.2) may not be susceptible to spikes 208, but may fail to closely fitthe intensity measurements in a neighborhood of a gap (e.g., gap 212).For example, the graph 216 of f(x) obtained from the minimizationoperation (2.1) closely fits the intensity readings b_(i) in aneighborhood of the spike 208 since readings adjacent to the spikeconstrain the maximization from entering the spiked region. However, agap (e.g., gap 212) may constrain the maximization from closely fittingthe intensities in a neighborhood of the gap as shown in FIG. 2.

The minimization and maximization operations (2.1) and (2.2) may beperformed using linear programming techniques. Accordingly, to linearizethe constraints of (2.1) and (2.2), the ratio of measurements adjacentto the “center” of each peak can be calculated for a given resolutionusing the equation for g(x) above at (1.2) where$a = \frac{resolution}{2\sqrt{\ln \quad 2}}$

and where the resolution is quantified as the width of the distributionat half height since the width of the distribution at half height isindependent of the peak height as one skilled in the art willunderstand. Note that the term “center” as used herein denotes apoint/region corresponding to an output measurement/reading of the waveform outputting process/application wherein the point/region has apredetermined characteristic such as being a local extreme value, havingchanges in the second derivative between positive and negative, and/orzero first derivative, or the point at which the integral reaches anextremum (often this is done since integrating smoothes derivatives ofnoisy data that are difficult to interpret.

Note that the objective function is simply the sum of the coefficientsc; in the equation for f(x) above, wherein each of the c_(i)'s representthe value of the maximum height of the corresponding discrete wave form.For the linearized problem the objective function is similar. Since itis assumed that the discrete wave forms are centered on integers, theobjective function is the sum of the c_(i) values corresponding to thediscrete wave form centers. Depending on which problem we are solving,we will either minimize or maximize this sum.

The constraints of the program are derived directly from the input datadescribed above. For the maximization problem we restrict f({overscore(x)}) to be less than or equal to {overscore (b)}. For the linearizedmaximization problem, this translates to a weighted sum of adjacentpeaks being smaller than the observed {overscore (b)}. Similarly, forthe minimization problem, we restrict f({overscore (x)}) to be greaterthan or equal to {overscore (b)}, with the linearized minimizationproblem translating to a weighted sum of adjacent peaks being largerthan the observed {overscore (b)}. Note that the weights for theweighted sum are derived for instantiations of g(x) of (1.2) with aknown resolution. FIG. 3 may be used to illustrate how the constraints(2.1) and (2.2) form inequalities, which may be converted to aconstraint matrix A as described hereinbelow.

If we assume that there is no cross-talk (meaning no contribution of onediscrete wave form to another discrete wave form), then the value off(x_(i)) can be taken to be either less than or equal to, or greaterthan or equal to the value of b_(i). Using the notation of FIG. 3 as anexample would take the form for the maximization operation (2.2):

f(x₃) ≦ b₃ which linearizes to h · c₃ ≦ b₃ f(x₅) ≦ b₅ h · c₅ ≦ b₅ f(x₇)≦ b₇ h · c₇ ≦ b₇

Accordingly, as one skilled in the art will understand, an instance ofthe constraint matrix A can be provided, wherein values for the c_(i)can be obtained by obtaining a solution to the following matrixequation: A{overscore (x)}<={overscore (b)}. Moreover, when h=1, theassociated constraint matrix A takes the following form (the x_(i)'s andthe ci's are for notational convenience only and are not part of thematrix A), wherein corresponding columns for c₁, c₂, c₄, c₆, c₈, and c₉are each the zero vector:

c₃ c₅ c₇ x₁ 0 0 0 x₂ 0 0 0 x₃ 1 0 0 x₄ 0 0 0 x₅ 0 1 0 x₆ 0 0 0 x₇ 0 0 1x₈ 0 0 0 x₉ 0 0 0

However, note that in the above “no cross talk” case, there is virtuallyno advantage as compared to using the raw spectrum measurements sincethe optimal values will be identical (or at most a scalar multiplethereof) to those of the original {overscore (b)} vector.

Augmentation of the constraints of (2.1) and (2.2) to includecontributions from adjacent discrete wave forms to other discrete waveforms is called cross-talk. If we restrict the region of influence ofeach discrete wave to only those discrete wave forms that are withinsome predetermined distance, then this type of cross talk is calledlimited cross-talk. In FIG. 3, it is clear that both the first andsecond discrete wave forms (304 and 308 respectively) have a significantinfluence on the value of x₅, while the first wave form 304 has verylittle, if any, influence on the discrete wave form 312. If we translatethese observations into a set of inequalities for the maximizationoperation (2.2) then we arrive at the following:

f(x₁) ≦ b₁ which linearizes to h₃ · c₃ ≦ b₁ f(x₂) ≦ b₂ h₂ · c₃ ≦ b₂f(x₃) ≦ b₃ h₁ · c₃ + h₃ · c₅ ≦ b₃ f(x₄) ≦ b₄ h₂ · c₃ + h₂ · c₅ ≦ b₄f(x₅) ≦ b₅ h₃ · c₃ + h₁ · c₅ + h₃ · c₇ ≦ b₅ f(x₆) ≦ b₆ h₂ · c₅ + h₂ · c₇≦ b₆ f(x₇) ≦ b₇ h₃ · c₅ + h₁ · c₇ ≦ b₇ f(x₈) ≦ b₈ h₂ · c₇ ≦ b₈ f(x₉) ≦b₉ h₃ · c₇ ≦ b₉

where${h_{1} = {g(0)}},{h_{2} = {g\left( {\pm \frac{1}{2}} \right)}},{h_{3} = {g\left( {\pm 1} \right)}}$

for g(x) as defined in (1.2).

The associated constraint matrix A would take the following form,wherein all other columns for c_(i)'s not presented are the zero vector:

c₃ c₅ c₇ x₁ h₃ 0 0 x₂ h₂ 0 0 x₃ h₁ h₃ 0 x₄ h₂ h₂ 0 x₅ h₃ h₁ h₃ x₆ 0 h₂h₂ x₇ 0 h₃ h₁ x₈ 0 0 h₂ x₉ 0 0 h₃

If instead of limited cross talk, every discrete wave form instance canat least potentially influence every other discrete wave form instancein the composite wave form, then this type of cross talk is known ascontinuous cross talk. In particular, a very high and broad discretewave form instance I_(i) can have influence on a low intensity discretewave form instance I_(j) that is spaced apart from I_(i) by a relativelylarge number of samples x_(k). As in the limited cross talk case above,the constraints of (2.1) and (2.2) for the non-linearized embodiment areidentical to the linearized version. However, when converting to thelinearized embodiment in the continuous cross talk case, every discretewave form instance is considered as contributing (at least potentially)to every other discrete wave form instance at every b_(i). Since theinequalities associated with the minimization operation (2.1) are merelythe reverse of the inequalities for the maximization operation (2.2)(i.e., “≧” instead of “≦”), only the equations for the maximizationoperation are provided:

f(x₁) ≦ b₁ which linearizes to h₃ · c₃ + h₅ · c₅ + h₇ · c₇ ≦ b₁ f(x₂) ≦b₂ h₂ · c₃ + h₄ · c₅ + h₆ · c₇ ≦ b₂ f(x₃) ≦ b₃ h₁ · c₃ + h₃ · c₅ + h₅ ·c₇ ≦ b₃ f(x₄) ≦ b₄ h₂ · c₃ + h₂ · c₅ + h₄ · c₇ ≦ b₄ f(x₅) ≦ b₅ h₃ · c₃ +h₁ · c₅ + h₃ · c₇ ≦ b₅ f(x₆) ≦ b₆ h₄ · c₃ + h₂ · c₅ + h₂ · c₇ ≦ b₆ f(x₇)≦ b₇ h₅ · c₃ + h₃ · c₅ + h₁ · c₇ ≦ b₇ f(x₈) ≦ b₈ h₆ · c₃ + h₄ · c₅ + h₂· c₇ ≦ b₈ f(x₉) ≦ b₉ h₇ · c₃ + h₅ · c₅ + h₃ · c₇ ≦ b₉

where${h_{1} = {g(0)}},{h_{2} = {g\left( {\pm \frac{1}{2}} \right)}},{h_{3} = {g\left( {\pm 1} \right)}},{h_{4} = {g\left( {{\pm 1}\frac{1}{2}} \right)}},{h_{5} = {g\left( {\pm 2} \right)}},{h_{6} = {g\left( {{\pm 2}\frac{1}{2}} \right)}},{h_{7} = {{g\left( {\pm 3} \right)}.}}$

The associated constraint matrix A would take the following form,wherein all other columns for c_(i)'s not presented are the zero vector:

c₃ c₅ c₇ x₁ h₃ h₅ h₇ x₂ h₂ h₄ h₆ x₃ h₁ h₃ h₅ x₄ h₂ h₂ h₄ x₅ h₃ h₁ h₃ x₆h₄ h₂ h₂ x₇ h₅ h₃ h₁ x₈ h₆ h₄ h₂ x₉ h₇ h₅ h₃

Accordingly, the constraint matrix A becomes much more dense as we allowthe influence of each peak instance to affect all other peak instances.In particular, the matrix A is not diagonally banded about the diagonalof A, instead at least one entire column has entirely non-zero entries.

In one embodiment of the present invention for performing massspectrometry, the operations of (2.1) and (2.2) are implemented using anoptimization algorithm based on the simplex method; in particular, theoptimization algorithm used to provide solutions to the operation of(2.1) and (2.2) was the simplex algorithm in the software packageMathematica® 4.0 of Wolfram Research Inc., 100 Trade Center Drive,Champaign, Ill. 61820-7237 USA. Note that the standard form of suchoptimization problems that Mathematica uses is: minimize z=Σc_(i)·x_(i)subject to A{overscore (x)}≧{overscore (b)} with {overscore (x)}≧0.

Note that the accompanying APPENDIX provided herewith providesMathematica® 4.0 programs embodying the minimization and maximizationoperations (2.1) and (2.2) respectively. Further note that each programsegment provided in the APPENDIX is immediately followed by an exampleoutput from the program segment to assist the reader in understandingthe program segment.

A description of the tests performed on the above described embodimentusing the simplex method is presented hereinbelow. However, the varioussets of test data used to evaluate the simplex based embodiment ofoperations (2.1) and (2.2) will be discussed first. Each set of testdata for a simulated composite wave form was created by summing aplurality of Gaussian curves (i.e., discrete wave forms as shown byf({overscore (x)}) in (1.1) above). The resulting sum was sampled at arate of five equidistant points per unit in the input ({overscore (x)})range. Random noise was then added to each corresponding intensity valueto more closely mimic real data (i.e., data from a non-simulated signalgenerating process). The random noise was calculated as some percentage(e.g., 5%) of the intensity at each x_(i). There were several differenttypes of test data sets generated, each type was designed to test adifferent aspect of the performance of the present invention. For eachof the test sets, three peaks were simulated, centered at units 2, 3,and 4 with intensities of 2, 3, and 4 respectively. An idealized testset with no random noise and a very high resolution (e.g., 0.5) is shownin FIG. 4.

Low resolution data often occurs as a result of hardware limitations. Totest low resolution data where intrusion of one discrete wave forminstance on the next is considerable, the width at half maximumintensity known in the art as Full-Width-Half-Max (FWHM) was set to 1.5units. This translates to approximately 30% of the intensity of one peakstill being present in an adjacent peak's maximal value. A test sethaving this low resolution characteristic is shown in FIG. 5 and will bereferred to as TEST₁.

When one peak is much stronger than an adjacent peak, often the smallerpeak is masked in a composite wave form instance so that it is notdetectable. In addition, the contribution of the larger peak may extendnoticeably to non-adjacent weak peaks. To simulate such a condition in atest data set, the resolution was set to 0.7 FWHM units (wherein therewas only 0.35% intrusion of one peak on the next), and the intensity ofthe middle peak will be changed from 3 to 100. The test set is shown inFIG. 6 and will be referred to as TEST₂. Note that 604 shows a reducedsize full copy of the graph 608 of the composite wave form instance. Asshown in this figure, the adjacent peaks at 2 and 4 x-units aresubstantially masked by the large peak at 3 units.

The test set shown in FIG. 7 includes a spike of 2 intensity units addedto b₈. The resolution is set to 0.5 FWHM units (only 0.0015% intrusionof one peak to the next). This test set is referred to as TEST₃.

The test set shown in FIG. 8 includes a decrease of 2 intensity units (agap) at b₇ (the middle of the third peak). The resolution for this dataset is set to 0.5 FWHM units (only 0.0015% intrusion of one peak to thenext). This test set is referred to as TEST₄.

The test set shown in FIG. 9 includes both poor resolution and apositive noise spike (which in mass spectrometry is more likely than anegative spike). Accordingly, this test set tests the performance of thelinear programming embodiment of the present invention under conditionsthat more closely mimic reality. The resolution will be set to 1.5 FWHM(30% intrusion of one peak to the next) and a positive noise spike of 2will be added to b₈. This test set is referred to as TEST₅.

Implementation of LP on the Test Data Sets

In this section, the results of both the minimization and maximizationoperations of (2.1) and (2.2) described. The results are provided forthe limited and continuous cross-talk models only since the model havingno cross-talk is substantially identical to the original input data.FIGS. 10 through 19 graphically illustrate the results of applyingminimization and maximization operations to the data test sets TEST₁through TEST₅. In each of the graphs, the f(x) generated according to(1.1) from the minimization operation (2.1) is labeled “MIN”, while thef(x) generated according to (1.1) from maximization operation (2.2) arelabeled “MAX”. The dashed graphs represent the reconstructeddistribution of instances of discrete wave forms obtained from thec_(i)'s determined by the minimization and maximization operations. Inparticular, the dashed graphs labeled P_(MIN) are the instances of thediscrete wave forms obtained from decomposing the MIN curve, and thegraphs labeled P_(MAX) are the instances of the discrete wave formsobtained from decomposing the MAX curve.

In general, upon viewing all the FIGS. 10 through 19, the continuouscross-talk fits the initial input data of each test data set onlyslightly better than the limited cross-talk with the maximization andminimization operations fitting to the input data substantially equallyin the continuous cross-talk and limited cross-talk cases. Each of theFIGS. 10 through 19 will now be discussed.

FIGS. 10 and 11 were obtained from the test data set TEST₁ for thelimited and continuous cross-talk respectively, wherein for the limitedcross-talk, each discrete wave form is limited to have effect at mostone mass unit away. The difference between the fits of the correspondingMIN and MAX is substantially constant and corresponds to the 5% randomvariation in the data.

FIGS. 12 and 13 were obtained from the test data set TEST₂ for thelimited and continuous cross-talk respectively, wherein for the limitedcross-talk, each discrete wave form is limited to have effect at mostone mass unit away. The continuous cross-talk performs significantlybetter than the limited cross-talk with the maximization andminimization operations performing substantially equally in thecontinuous and the maximization producing better results in the limitedcross-talk case. The difference between the fits is relatively large inthe right most P_(MIN) peak and the right most P_(MAX) peak in thelimited cross-talk case (FIG. 12).

FIGS. 14 and 15 were obtained from the test data set TEST₃ for thelimited and continuous cross-talk respectively, wherein for the limitedcross-talk, each discrete wave form is limited to have effect at mostone mass unit away. In both the limited and continuous cases, themaximization operation performs significantly better than theminimization operation. The primary difference between the fits of f(x)determined by the minimization operation (2.1) and the maximizationoperation (2.2) is illustrated by the difference in the right mostP_(MIN) peak and the right most P_(MAX) peak.

FIGS. 16 and 17 were obtained from the test data set TEST₄ for thelimited and continuous cross-talk respectively, wherein for the limitedcross-talk, each discrete wave form is limited to have effect at mostone mass unit away. In both the limited and continuous cases, theminimization operation performs significantly better than themaximization operation. The primary difference between the fits of f(x)determined by the minimization operation (2.1) and the maximizationoperation (2.2) is illustrated by the difference in the right mostportion of each graph.

FIGS. 18 and 19 were obtained from the test data set TEST₅ for thelimited and continuous cross-talk respectively, wherein for the limitedcross-talk, each discrete wave form is limited to have effect at mostone mass unit away. In both the limited and continuous cases, themaximization operation performs significantly better than theminimization operation. The difference between the fits of the MIN andMAX is relatively large at the left most and right most pair of P_(MIN)and P_(MAX) for the limited cross-talk case, but only at the right mostpair of P_(MIN) and P_(MAX) for the continuous cross-talk case.

The graphical results shown in FIGS. 10-19 illustrate that thecontinuous cross-talk models always outperformed the limited cross-talkmodels. Moreover, it can be empirically demonstrated that in someapplications of mass spectrometry (e.g., determining the purity of aparticular substance) that cross-talk can occur at distances furtherthan 1 mass unit away from a peak and therefore necessitate the use ofcross-talk which is further-reaching than the limited cross talk case.In general it should typically not be necessary to utilize a continuouscross-talk model, but instead a model wherein the cross-talk extends farenough to where the contribution of one peak to another is less than,e.g., 0.001% to 1% of the smaller peak's value. Additionally, from testsperformed by the Applicant for mass spectrometry, the maximizationoperation has proved to be more accurate in representing the input massspectrometry data, except for the noise gap problem where theminimization operation was clearly better. Since gap noise is more rare,in general the maximization operation algorithm is preferred for massspectrometry analysis. However, for other applications where gap noiseis more dominant, the minimization operation may be preferred. Forexample, in such signal processing applications as absorbancespectroscopy the minimization operation (2.1) may be preferred as oneskilled in the art will understand.

It is also an aspect of the present invention to utilize both theminimization and maximization operations together on the same input dataset. For example, there are various techniques in the art for reliablydetermining the centers (e.g., extreme points, or more generally, apredetermined region of a wave form) of the discrete wave forms. Inparticular, such points/regions may be determined using, e.g.,derivatives, and weighted averaging as one skilled in the art willunderstand. Accordingly, a pair P_(MIN) and P_(MAX) can be generated foreach center. Accordingly, for each discrete wave form center, thecorresponding pair of discrete wave forms P_(MIN) and P_(MAX) may beused to obtain a resulting discrete wave form. In particular, for eachcenter, a resulting discrete wave form may be derived:

(a) As a weighted combination of the corresponding pair of discrete waveforms, or

(b) By selecting one of the discrete wave forms P_(MIN) and P_(MAX) ofthe corresponding pair.

In either case, a determination can be made as to which one of thediscrete wave forms P_(MIN) and P_(MAX) of the corresponding pairappears to determine the composite function, such as f(x) of (1.1), thatbests the original input data (x_(i), b_(i)). Moreover, there arenumerous techniques for making such a determination. In one embodimentwherein a linear programming technique is used to linearly approximatef(x) for both the minimization and maximization operations as describedhereinabove, such a determination can be made for a center x₀ (havingP_(MIN(0)) and P_(MAX(0)) as its corresponding discrete wave forms) byusing x_(i) of an input pair (x_(x), b_(i)) relatively close to x₀ as areplacement center for x₀ and subsequently re-perform the minimizationand maximization operations to obtain new versions of the discrete waveforms P_(MIN(i)) and P_(MAX(i)). Thus, if the original center x₀happened to correspond to a spike or a gap, then by using the nearbyvalue x_(i) as a replacement center (presumably not a spike or a gap),one of the newly derived corresponding discrete wave forms P_(MIN(i))and P_(MAX(i)) will likely change amplitude dramatically from that ofthe respective P_(MIN(0)) and P_(MAX(0)), while the difference betweenthe other one of P_(MIN(i)) and P_(MAX(i)) and its respective P_(MIN(0))and P_(MAX(0)) will change relatively little. Thus, the one ofP_(MIN(0)) and P_(MAX(0)) that deviates least from its newer version isdetermined to be most indicative of the true non-noise reading at x₀ forthe application (e.g., mass spectrometry assays). Accordingly, theweights for the discrete wave forms at x₀ referred to in (a) above maybe determined to depend inversely on the differences between thecorresponding original and new off center versions of the discrete waveforms. For instance, the resulting discrete wave form for x₀ could be,e.g., a weighting function may take the results of the minimizationoperation P_(MIN(0)) and the results of the maximization problemP_(MAX(0)) and combine them so that the weighting function is one of:

P=(P _(MIN(0)) +P _(MAX(0)))/2;  (i)

P=((P _(MIN(0)){circumflex over ( )}2+P _(MAX(0)){circumflex over ()}2){circumflex over ( )}1/2)/2;  (ii)

or

P=((P _(MIN(0)){circumflex over ( )}3+P _(MAX(0)){circumflex over ()}3){circumflex over ( )}1/3)/2.  (iii)

Alternatively, in (b) immediately above, the one of P_(MIN(0)) andP_(MAX(0)) that deviates least from its newer version is determined tobe most indicative of the true non-noise reading at x₀ and is thereforeselected as the discrete wave form for x₀.

FIGS. 20A through 20C is a high level flowchart of the steps performedby the present invention for decomposing a composite wave form into aplurality of discrete wave forms. Note, that the steps of this flowchartare preferably performed within an appropriately programmed computerthat receives data samples (e.g., of the form (x_(i), b_(i)) asdescribed hereinabove). Moreover, such data samples may be obtained fromvarious types instruments and for various types of applications such as:

(3.1) Mass spectrometry;

(3.2) Absorption/Transmission/Emission spectroscopy;

(3.3) Fluorescence Spectroscopy;

(3.4) Raman Spectroscopy;

(3.5) Digital Image enhancement (i.e. de-blurring and de-noising);and/or

(3.6) Sound Enhancement.

Note that the applications described here will be more fully describedherein below.

The flowchart of FIGS. 20A through 20C will now be described. In step2004 a determination is made as to what distribution functional form Gis appropriate for representing the discrete wave forms into which acomposite wave form is to be decomposed. Depending upon the wave formoutputting application from which data samples are received, variousdistribution functional forms G may be used such as Gaussian, Bessel,characteristic (step) functions, and sigmoid. In particular, for thesignal processing applications (3.1) through (3.6) described above, thefollowing functional forms G may be used:

(a) Gaussian;

(b) Multivariate Gaussian;

(c) Bessel;

(d) Hat functions;

(e) Wavelets; and/or

(f) any function that can be parameterized and made to be in the form ofan independent spanning set.

In step 2008, determine the values for the parameters that may affectthe decomposition of the composite wave form into its discrete waveforms. In particular, the following values are determined: (a) thedomain (D) from which values are input to the wave form outputtingapplication for obtaining corresponding readings, and (b) the samplingfrequency (or frequencies) to be used in determining a collection X ofinput values x_(i)εD, 1<=i₀<=i<k for some integers i₀ and k, to input tothe wave form outputting application for thereby obtaining correspondingapplication readings b_(i). 1<=i₀<=i<=k. Additionally, note that theresolution (R) of the of the wave form outputting application may bedesirable to obtain, since, e.g., resolution is used to determine theentries (h_(i)) in the matrix A described in the linear programmingembodiment hereinabove in that resolution is used in defining the term“a” in the function g(x) of (1.2). Note that steps 2004 and 2008 neednot be performed within the computational system that subsequent stepsof FIGS. 20A through 20C are performed. In fact, steps 2004 and 2008 maybe performed prior to activation of the wave form outputting applicationand may be determined by empirical tests and/or manually. For example,in performing various mass spectrometry assays the functional form G ofstep 2004 may be well known, and the parameter values determined in step2008 may have been previously set in the mass spectrometer.

In step 2012, the present step receives, from the wave form outputtingapplication, a collection B of sample measurements or readings of thecomposite wave form at sampling frequency (or frequencies) F, whereinB={b_(i) wherein b_(i) is a measurement or reading of the composite waveform at x_(i)εX}. Note that the data received from the wave formoutputting application may in the form of data structures representingeach of the pairs (x_(i), b_(i)). Further note that preferably F (oreach such frequency) is greater than R. Note, that subsequent steps inFIGS. 20A through 20C assume a correspondence or association betweeneach b_(i) and x_(i)ε X. Such correspondence may be represented by datastructures that are received in the present step; e.g., data structuresrepresenting pairs (x_(i), b_(i)) x_(i) ε X,

Note that the order of steps of 2004 through 2012 may not be importantin at least some applications of this flowchart.

In step 2016, a collection CNTR of values from X are determined that areexpected to be the centers (or another identifiable point) of each ofthe discrete wave forms to be determined. Note that in many wave formoutputting applications (e.g., mass spectrometry), the values of CNTRcorrespond to extreme measurements or readings b_(i) the values of CNTRmay be determined by various techniques such as the following:

(5.1) numerical differentiation (upwind, downwind, central difference)and finding zeros;

(5.2) second derivative to find where the slope changes from positive tonegative; or

(5.3) smoothing then choosing maximal values.

In step 2020, for each x_(i) in CNTR, determine a likely cross-talksupport CTS_(i) of an instantiation I_(i) of the distribution functionalform G having an unknown extreme measurement or reading at x_(i),wherein CTS_(i) is represented a series of contiguous values {x_(j(i))where M_(1i)<=j(i)<=M_(2i) with i₀<=M_(1i)<=M_(2i)<=k and x_(i) is oneof the x_(j)}. with M_(1i) and M_(2i) corresponding to the limits on therange in D for the cross-talk support of I_(i). Note that in some waveform outputting applications, such as mass spectrometry, the cross-talksupport may be different for different x_(i) since the resolution of amass spectrometer may vary over D. Note, that this step need notnecessarily be performed after step 2016 as indicated in the flowchartof FIGS. 20A through 20C. In fact, step 2020 may be performed earlier,but after step 2004.

In step 2024, for each cross-talk support CTS_(i), determine for eachx_(j) (of X) in CTS_(i), M_(1i)<=j<=M_(2i), a corresponding expressionwhich represents at least an approximation of I_(i) at x_(j). Note thateach of the expressions may be function h_(ij)(C_(i)), whereinh_(ij)(C_(i)) corresponds to an approximation of the intensity oramplitude of I_(i) at x_(j), and wherein C_(i) is the unknownintensity/amplitude of I_(i) at x_(i). In one embodiment of the presentinvention, the expressions h_(ij)(C_(i)) are determined using linearprogramming techniques, e.g., as described hereinabove in the linearprogramming examples given.

In step 2028, let AC be the vector$\left\lbrack {\sum\limits_{j = i_{0}}^{k}{h_{ij}\left( C_{i} \right)}} \right\rbrack,{{i_{0}<=i<={k\quad {where}\quad {h_{ij}\left( C_{i} \right)}}} = 0}$

for such terms not otherwise define in step 2024. Now maximize theobjective function $\sum\limits_{j = i_{0}}^{k}C_{i}$

subject to AC<=B, thereby obtaining the vector C_(MAX)=[c_((MAX,i))],wherein c_((MAX,i)) is an approximation of the intensity/amplitude atx_(i) OF I_(i). Note that this step may be performed using a linearprogramming technique such as the simplex method as one skilled in theart will understand. Further note that this step corresponds toperforming the maximization operation (2.2).

In step 2032, minimize the objective function$\sum\limits_{j = i_{0}}^{k}C_{i}$

where i₀<=i<=k, again subject to AC>=B as described in step 2028 toobtain the vector C_(MIN)=[c_((MIN,i))], wherein c_((MIN,i)) is anapproximation of the intensity or amplitude at x_(i) of I_(i). Note thatthis step may also be performed using a linear programming techniquesuch as the simplex method as one skilled in the art will understand.Further note that this step corresponds to performing the minimizationoperation (2.1).

In step 2036, a determination is made as to which of various values bestrepresents a measurement/reading that would have been output by the waveform outputting process/application if there are were substantially nonoise (e.g., spikes and/or gaps). There are numerous techniques forperforming such a determination and such various determinations arewithin the scope of the present invention. Step 2036 illustrates, at ahigh level, one such technique for determining an appropriate noisereduced/eliminated output b_(i) for each center x_(i) in CNTR. That is,for each x_(i) in CNTR the following substeps (i) through (iv) followingare performed:

(i) replace b_(i) with (c_((MAX,i))−c_((MIN,i)))/2 (or some otherappropriately chosen value between (or including) c_((MAX,i)), andc_((MIN,i)));

(ii) recompute both c_((MAX,i)), c_((MIN,i)) using the replacement valuefor b_(i), thereby obtaining values for nc_((MAX,i)), and nc_((MIN,i));

(iii) determine how much each of the new nc_((MAX,i)), and nc_((MIN,i))changed from its corresponding original value, c_((MAX,i)), andc_((MIN,i)) respectively. Note, that if the original b_(i) correspondsto a spike, then the new b_(i) also corresponds to a spike (albeit alesser one), and nc_((MIN,i)) is expected to move closer to thereplacement value of b_(i) provided in substep (i) than nc_((MAX,i)).Alternatively, if the original b_(i) corresponds to a gap, then thereplacement b_(i) also corresponds to a gap (albeit a lesser one), andnc_((MAX,i)) is expected to move closer to the replacement value ofb_(i) provided in substep (i) than nc_((MIN,i)). Note that there areadditional techniques that can be used in this step for determiningwhich of c_((MAX,i)), and c_((MIN,i)) provide a more accurate (i.e.,less noisy) value than b_(i). For example, the nc_((MAX,i)), andnc_((MIN,i)) farthest from the replacement value for b_(i) may beindicative of the more accurate one of c_((MAX,i)), and c_((MIN,i));

(iv) determine a new intensity/amplitude nc_(i) for I_(i) wherein thenew intensity is a function that is dependent upon the result fromsubstep (iii). More particularly, nc_(i) may be whichever one of theoriginal c_((MAX,i)), c_((MIN,i)) whose corresponding new valuesnc_((MAX,i)), nc_((MIN,i)) respectively moved the least toward thereplacement value of b_(i). Alternatively, the new intensity may be aweighted sum of the original c_((MAX,i)), c_((MIN,i)) wherein suchweights are, e.g., inversely related to the relative changes betweenΔc_((MAX,i))=ABSOLUTE VALUE OF [c_((MAX,i))−nc_((MAX,i))], andΔc_((MIN,i))=ABSOLUTE VALUE OF [c_((MIN,i))−nc_((MIN,i))].

In step 2040, for each instance I_(i) of a discrete wave form obtained,output data indicative of the instance's center and its correspondingintensity/amplitude for thereby identifying a physical characteristic ofthe item from which the composite wave form was obtained. Note thatsubsequently, depending on the use and interpretation supplied to theoutput data various post processing steps and/or results may be achievedby different embodiments of the present invention. Such post processingsteps and/or results are described hereinbelow in the context ofdifferent embodiments of the present invention.

Embodiments of the present invention can be used to process signals fromvarious wave form outputting applications. The following descriptions(A) through (C) following further describe such wave form outputtingapplications to which an embodiment of the present invention can beapplied:

(A) Mass Spectroscopy

FIG. 21 is a diagram illustrating the components of a mass spectrometrysystem 2100 that uses an embodiment of the present invention fordetermining at least one of: (a) an identity of a compound or material;(b) a concentration or purity of a compound or material; (c) a molecularweight of a compound or material; and (d) a molecular structure of acompound or material. The mass spectrometry system 2100 includes asample input device 2104 in which a sample of the compound(s) ormaterial(s) to be assayed resides when intensity readings ormeasurements are being taken by the mass spectrometer 2108, wherein suchreadings or measurements represent a composite wave form. Note that thesample input device 2104 may only house the sample(s) being assayedmomentarily; e.g., a gas or liquid may flow therethrough as the readingsare taken by the mass spectrometer 2108. Subsequently, the readings ormeasurements, together with their corresponding sampling input values inthe mass spectrometer sampling range (e.g., the mass to charge ratiorange), for the composite wave form are output to the noisereduction/resolution enhancement engine 2112 which is an operativeembodiment of the flowchart of FIGS. 20A through 20C. The noisereduction/resolution enhancement engine 2112 can reside in a computerdistinct from the mass spectrometer 2108. Moreover, note that the noisereduction/resolution enhancement engine 2112 may utilize the linearprogramming methods described hereinabove for operationalizing theflowchart of FIGS. 20A through 20C. However, non-linear techniques mayalso be used to operationalize FIGS. 20A through 20C in the noisereduction/resolution enhancement engine 2112. The output from the noisereduction/resolution enhancement engine 2112 is, e.g., for each discretewave form obtained from the composite wave form, a center of thediscrete wave form and a corresponding intensity/amplitude. This outputis supplied to a correlation engine 2116 which performs a correlation ora most likely match/similarity of the engine 2116 output with date forone or more discrete wave forms of known compounds residing in themolecular database 2120. Thus, the correlation engine 2116 outputs amost likely identity, concentration, molecular weight, and/or molecularstructure of the assayed sample to at least one of a display and/or astorage device (not shown), wherein the output may be used for furtheranalysis or for presenting to a user. Note that such a correlationengine 2116 may determine the correlation or most likelymatch/similarity by statistical techniques, hierarchical techniques, andtechniques that employ some measure of proximity to a known compound ormixture as one skilled in the art will understand.

(B) Light spectroscopy

Often in light spectroscopy one is attempting to measure discrete lightbands that correspond to a mode of the compound that one is attemptingto measure. These discrete light bands will appear as peaks that may beGaussian or Bessel type functions. As in mass spectrometry, the systemsthat measure these light bands suffer from noise and limited resolution.In particular, absorbance spectroscopy is highly susceptible to noisegap problems while transmission spectroscopy is inherently susceptibleto noise spike problems. Resolution is usually determined by thecharacteristics of the system such as entrance slit width, focal length,quality of the optics, and proper alignment.

One light spectroscopy application for which the present invention isparticularly useful is Raman spectroscopy which measures the shifts in alaser light line. The spectra from this type of application often appearas peaks each corresponding to a mode of the sample being assayed.

Another light spectroscopy application for which the present inventionis useful is fluorescence spectroscopy, wherein a sample to be assayedis excited with a discrete light source, which the sample absorbs thenre-emits the light at lower energy (longer wavelength). The intensityand wavelengths of the emitted light are often characteristic of thesubstance(s) in the sample and can often be used to determineconcentration, identity and even structure of the sample.

In light spectroscopy applications the the present invention can be usedto generate more accurate spectra which can be used to correlate withconcentration, identification and structure determination.

FIG. 22 is a diagram illustrating the components of a light spectrometrysystem 2200 that uses light spectrometer 2204 together with anembodiment of the present invention for determining at least one of: (a)an identity of a compound or material; (b) a concentration or purity ofa compound or material; and (c) a molecular structure of a compound ormaterial. The light spectrometry system 2200 includes a light source2202 for generating light directed toward a sample chamber 2208 in whicha sample of the compound(s) or material(s) to be assayed resides whenlight band readings or measurements are being taken by the lightspectrometer 2204, wherein such readings or measurements represent acomposite wave form. Subsequently, the readings or measurements,together with their corresponding sampling input values in the lightspectrometer sampling range (e.g., the range of light frequencies fromwhich the readings are to be obtained), for the composite wave form areoutput to the noise reduction/resolution enhancement engine 2212 whichis an operative embodiment of the flowchart of FIGS. 20A through 20C.The noise reduction/resolution enhancement engine 2212 can reside in acomputer distinct from the light spectrometer 2204. Moreover, note thatthe noise reduction/resolution enhancement engine 2212 may utilize thelinear programming methods described hereinabove for operationalizingthe flowchart of FIGS. 20A through 20C. However, non-linear techniquesmay also be used to operationalize FIGS. 20A through 20C in the noisereduction/resolution enhancement engine 2212. The output from the noisereduction/resolution enhancement engine 2212 is, e.g., for each discretewave form obtained from the composite wave form, a center of thediscrete wave form and a corresponding light band amplitude. This outputis supplied to a correlation engine 2216 which performs a correlation ormost likely match/similarity of the engine 2212 output with date for oneor more discrete wave forms of known compounds residing in the moleculardatabase 2216. Thus, the correlation engine 2216 outputs a most likelyidentity, concentration, and/or molecular structure of the assayedsample to at least one of a display and/or a storage device (not shown),wherein the output may be used for further analysis or for presenting toa user as one skilled in the art will understand. Note that such acorrelation engine 2212 may determine the correlation or most likelymatch/similarity by statistical techniques, hierarchical techniques, andtechniques that employ some measure of proximity to a known compound ormixture as one skilled in the art will understand.

(C) Image enhancement

Often digital images (especially those taken with a digital camera orscanner) suffer from noise and poor focus. The result is an image thathas stray noise and a blurred appearance. A technique that couldmitigate noise and de-blur an image would be very desirable since itwould improve the quality of digital images and could render otherwiseunusable images as useable. Often poor focus has the effect of aGaussian blur which affects the clarity of the image. To obtain a matrixof intensity values that represents such an image, the samelinearization techniques as described hereinabove for the presentinvention can be applied so that the constraint matrix A has as manyrows as there are pixels, and where the {overscore (b)} vector is theoriginal image as one skilled in the art will understand.

FIG. 23 is a diagram illustrating the components of an image enhancementsystem 2300 that uses a digital camera 2304 together with an embodimentof the present invention for enhancing digital images output by thedigital camera In particular, this embodiment of the present inventionincludes a noise reduction/resolution enhancement engine 2308 that canreside in a computer distinct from the digital camera 2304. Moreover,note that the noise reduction/resolution enhancement engine 2308 mayutilize the linear programming methods described hereinabove foroperationalizing the flowchart of FIGS. 20A through 20C. However,non-linear techniques may also be used to operationalize FIGS. 20Athrough 20C in the noise reduction/resolution enhancement engine 2308.The noise reduction/resolution enhancement engine 2308 receives, foreach pixel of the image output by the digital camera 2304, color data.Such color data can be noise reduced (e.g., deblurred) at each pixel by,e.g., considering the color data as a composite wave form anddecomposing the composite wave form to obtain the primary (i.e., highestamplitude) color(s) for the pixel, and subsequently the combination ofthese primary colors with their amplitudes as the enhanced color of thepixel that is output to at least one of a display and/or a storagedevice (not shown), wherein the output may be used for further analysisor for presenting to a user.

(D) Sound enhancement

Sound that has been digitized often suffers from noise. An techniquethat reduces this noise can make the sound more appealing and/orsuccinct. Sound data can be stored in several ways. In a pure form, itis a composite of many frequencies that are summed together to give acomposite waveform of amplitudes. This waveform is sampled at some rate,and can often have noise introduced into this sampling. An embodiment ofthe present invention can be used to “fit” sine and cosine waves ofvarying frequency to recreate the waveform with less noise. Toaccomplish this a window of allowable frequencies and the number ofdistinct frequencies allowed is determined prior to decomposingcomposite wave forms representing the sound.

FIG. 24 is a diagram illustrating the components of an sound enhancementsystem 2400 that uses a digital recorder 2404 together with anembodiment of the present invention for enhancing digital audio dataoutput by the digital recorder In particular, this embodiment of thepresent invention includes a noise reduction/resolution enhancementengine 2408 that can reside in a computer distinct from the digitalrecorder 2404. Moreover, note that the noise reduction/resolutionenhancement engine 2408 may utilize the linear programming methodsdescribed hereinabove for operationalizing the flowchart of FIGS. 20Athrough 20C. However, non-linear techniques may also be used tooperationalize FIGS. 20A through 20C in the noise reduction/resolutionenhancement engine 2408. The noise reduction/resolution enhancementengine 2408 receives a plurality of audio segments (each having aplurality of digital audio data packets) output by the digital recorder2404. Each such audio packet can be noise reduced by e.g., consideringthe audio data in the packet as a composite wave form and decomposingthe composite wave form to obtain the primary (i.e., highest amplitude)one or more audio frequencies for the packet, and subsequently thecombination of these primary audio frequencies with their amplitudes asthe enhanced audio packet data that is output to at least one of adisplay, and/or a storage device (not shown), wherein the output may beused for further analysis or for presenting to a user.

It is important to note that each of the noise reduction/resolutionenhancement engines 2112, 2212, 2308 and 2408 may be accessed through acommunications network such as the Internet. Thus, the data samplesinput to these engines (by e.g., the mass spectrometer 108, the lightspectrometer 2204, the digital camera 2304 or the digital sound recorder2404) may be received through such a communications network, and theoutputs obtained from the noise reduction/resolution enhancement engines(or results derived from such outputs, e.g., via correlation engines2116 or 2216) can be transmitted back to the source of the original datasamples.

While various embodiments of the present invention have been describedin detail, it is apparent that further modifications and adaptations ofthe invention will occur to those skilled in the art. However, it is tobe expressly understood that such modifications and adaptations arewithin the spirit and scope of the present invention.

What is claimed is:
 1. A signal processing method for obtaining dataindicative of a plurality of discrete wave forms obtained from acomposite wave form, wherein the composite wave form is obtained from awave form outputting process A, comprising: first determining a discretewave form wherein a functional combination, f(x) of a plurality ofinstances of the discrete wave form approximates the composite wave formwhen x varies over a predetermined range; receiving, for each sampleinput x of a collection X having a plurality of sample inputs to thewave form outputting process corresponding signal amplitude data, b_(x),output from the wave form outputting application wherein a collection ofpairs (x, b_(x)), x in X is indicative of the composite wave form;second determining, for each instance, I, of the discrete wave forminstances, a portion, p, of a range of one or more of the sample inputs,wherein the portion p corresponds to a instantiation in I of apredetermined portion of the of the discrete wave form, and wherein pincludes at least one of the sample inputs x whose corresponding signalamplitude data, b_(x), is substantially an extreme amplitude for I;first obtaining a solution for minimizing a functional σ that isdependent on a plurality of unknown reduced noise amplitudes c_(p),where there is one of the c_(p) for at least one corresponding valuex_(p) from each of the portions p, wherein σ monotonically increaseswith increases in a value for each c_(p) when the value for c_(p) is ina range substantially bounded by the set {b_(x), x in X}, wherein saidminimizing is subject to the constraint: f(x)≧b_(x) for x in X, whereinsaid solution includes an amplitude value c_(p,MIN) for each said c_(p);second obtaining a solution for maximizing said functional σ, whereinsaid maximizing is subject to the constraint: f(x)≦b_(x) for x in X,wherein said solution includes an amplitude c_(p,MAX) for each saidc_(p); identifying, for each said c_(p), which of said c_(p,MIN) andsaid c_(p,MAX) has a lesser amount of noise; associating, for each saidc_(p), the corresponding value x_(p) and an amplitude value nc_(p),wherein c_(p,MAX)<=nc_(p)<=c_(p,MIN), and nc_(p) is at least as close tosaid one of said c_(p,MIN) and said c_(p,MAX) identified in saididentifying step as to the other of said c_(p,MIN) and said c_(p,MAX);comparing (A1) and (A2) following for determining a similaritytherebetween: (A1) a resulting collection of associations of nc_(p) andx_(p), and (A2) a predetermined collection C of associations, each saidassociation of C including an amplitude and a corresponding value of thepredetermined range; outputting a result indicative of one of (B1)through (B4) following, wherein said result is dependent upon asimilarity determined in said step of comparing: (B1) a purity of asubstance assayed by the wave form outputting process ; (B2) an identityof a substance assayed by the wave form outputting process ; (B3) anamount of a substance assayed by the wave form outputting process ; and(B4) a structure of a substance assayed by the wave form outputtingprocess .
 2. The method of claim 1, wherein said step of outputtingincludes outputting the purity of the substance assayed by the wave formoutputting application .
 3. The method of claim 1, wherein said step ofoutputting includes outputting the identity of the substance assayed bythe wave form outputting application .
 4. The method of claim 1, whereinsaid step of outputting includes outputting the amount of the substanceassayed by the wave form outputting application .
 5. The method of claim1, wherein said step of outputting includes outputting the structure ofthe substance assayed by the wave form outputting application .
 6. Themethod of claim 1, wherein the discrete wave is one of: (a) Gaussian;(b) Multivariate Gaussian; (c) Bessel; (d) Hat functions; and (e)Wavelets.
 7. The method of claim 1, wherein the discrete wave is afunction parameterized so that it forms an independent spanning set. 8.The method of claim 1, wherein said step of receiving includesoutputting said amplitude data b_(x) from one of: a mass spectrometer,and a light spectrometer.
 9. The method of claim 1, wherein said step ofsecond determining includes determining for each instance, I, of thediscrete wave form, a center point of I said portion p.
 10. The methodof claim 1, wherein said functional σ includes a summation of terms,w_(p)*c_(p) for each said unknown reduced noise amplitude c_(p) andw_(p)>=0.
 11. The method of claim 1, wherein said step of firstobtaining includes obtaining a matrix A wherein an entry a_(i,j) of Arepresents a cross-talk between discrete wave form instances I_(i) andI_(j).
 12. The method of claim 11, wherein said step of first obtainingincludes performing a linear programming technique for minimizing saidfunctional σ subject to the constraint: A{overscore (x)}≦{overscore(b)}, wherein {overscore (b)} is a vector whose entries include theb_(x) for x in X.
 13. The method of claim 1, wherein said secondobtaining includes obtaining a matrix A wherein an entry a_(i,j) of Arepresents a cross-talk between discrete wave form instances I_(i) andI_(j).
 14. The method of claim 13, wherein said step of first obtainingincludes performing a linear programming technique for minimizing saidfunctional σ subject to the constraint: A{overscore (x)}≧{overscore(b)}, wherein {overscore (b)} is a vector whose entries include theb_(x) for x in X.
 15. The method of claim 1, wherein said step ofidentifying includes: determining, for at least one said c_(p), adifferent value from both of c_(p,MIN) and c_(p,MAX) for the signalamplitude b_(x) _(p) for the sample input x_(p) corresponding to c_(p);and performing said steps of first obtaining and second obtaining againfor values nc_(p,MIN) and nc_(p,MAX) for said at least one c_(p);deriving a reduced noise amplitude value for said at least one c_(p)using said nc_(p,MIN) and nc_(p,MAX).
 16. The method of claim 15,wherein said step of deriving includes determining how much at least oneof said nc_(p,MIN) and nc_(p,MAX) varies, respectively, from one of: (a)said different value, and (b) said c_(p,MIN) and c_(p,MAX) for said atleast one c_(p).
 17. The method of claim 16, wherein said step ofderiving includes determining said reduced noise amplitude value forsaid at least one c_(p), wherein said reduced noise amplitude value iscloser to the one of said C_(p,MIN) and c_(p,MAX) whose correspondingvalue nc_(p,MIN) and nc_(p,MAX) moved a least amount toward saiddifferent value.
 18. A method for obtaining noise reduced data frommeasurements corresponding to each sample of a collection X having aplurality of data samples x, comprising: receiving signal amplitudedata, b_(x) for each of x in X, wherein X includes a plurality of valueswithin a predetermined range; selecting a subcollection CNTR of X;minimizing a functional σ that is dependent on a plurality of unknownreduced noise amplitudes c_(x), where there is one of the c_(x) for eachx in CNTR, and wherein σ monotonically increases with increases invalues for each c_(x) when the values for c_(x) are in the predeterminedrange; wherein said step of minimizing is subject to the constraint:f(x)≧b_(x) for x in X, wherein f(x) is a function for approximating thepairs (x, b_(x)) for x in X wherein a solution obtained from saidminimizing step includes an amplitude value c_(x,MIN) for each saidc_(x); maximizing said functional (subject to the constraint: f(x)≦b_(x)for x in X; wherein a solution to said maximizing step includes anamplitude c_(x,MAX) for each said c_(x); determining, for each saidc_(x), which of said c_(x,MIN) and said c_(x,MAX) includes a lesseramount of noise; obtaining, for each said c_(x), an amplitude valuenc_(x), wherein c_(x,MAX)<=nc_(x)<=c_(x,MIN), and nc_(x) is at least asclose to said one of said c_(x,MIN) and said c_(x,MAX) determined insaid determining step as to the other of said c_(x,MIN) and saidc_(x,MAX); outputting a result obtained using at least one nc_(x),wherein said result is indicative of one of: (i) a purity, identity,amount, or structure of a substance, wherein the collection X is anoutcome of an assay of the substance; (ii) data for one or more pixels,wherein the pixels are elements x of the collection X; and (iii) datafor one or more audio segments, wherein the segments are elements x ofthe collection X.
 19. The method of claim 18, wherein said result isindicative of a purity, identity, amount, or structure of a substance,wherein the collection X is an outcome of an assay of the substance. 20.The method of claim 18, wherein said result is indicative of data forone or more pixels, wherein the pixels are elements x of the collectionX.
 21. The method of claim 18, wherein said result is indicative datafor one or more audio segments, wherein the segments are elements x ofthe collection X.
 22. The method of claim 18, wherein said step ofdetermining includes: determining, for at least one said c_(x), adifferent value from both of c_(x,MIN) and c_(x,MAX) for the signalamplitude b_(x) corresponding to the data sample x corresponding toc_(x); and performing said steps of minimizing and maximizing again forobtaining values nc_(x,MIN) and nc_(x,MAX) for said at least one c_(x);deriving a reduced noise amplitude value for said at least one c_(x)using said nc_(x,MIN) and nc_(x,MAX).
 23. The method of claim 22,wherein said step of deriving includes determining how much each of saidnc_(x,MIN) and nc_(x,MAX) varies, respectively, from one of: (a) saiddifferent value, and (b) said c_(x,MIN) and c_(x,MAX) for said at leastone c_(x).
 24. An apparatus for reducing noise in measurementscorresponding to each value of a collection X having a plurality of datasamples x, comprising: a device for generating measurements b_(x) foreach x in X, wherein said device determines each said b_(x) from one of:an amplitude, and an intensity of a signal received from a sourceexternal to said device, wherein said collection X includes values xindicative of one of: (A1) a purity, identity, amount, or structure of asubstance, wherein the collection X is an outcome of an assay of thesubstance; (A2) data for one or more pixels, wherein the pixels areelements x of the collection X; and (A3) data for one or more audiosegments, wherein the segments are elements x of the collection X, anoise reduction/resolution enhancement engine for performing thefollowing steps (a) through (e): (a) selecting a subcollection CNTR ofX; (b) minimizing a functional σ that is dependent on a plurality ofunknown reduced noise amplitudes c_(x), where there is one of the c_(x)for each x in CNTR, and wherein σ monotonically increases with increasesin values for each c_(x) when the values for c_(x) are in thepredetermined range; wherein said step of minimizing is subject to theconstraint: f(x)≧b_(x) for x in X, wherein f(x) is a function forapproximating the pairs (x, b_(x)) for x in X wherein a solutionobtained from said minimizing step includes an amplitude value c_(x,MIN)for each said c_(x); (c) maximizing said functional σ subject to theconstraint: f(x)≦b_(x) for x in X; wherein a solution to said maximizingstep includes an amplitude c_(x,MAX) for each said c_(x); (d)determining, for each said c_(x), which of said c_(x,MIN) and saidc_(x,MAX) includes a lesser amount of noise; (e) obtaining, for eachsaid c_(x), an amplitude value nc_(x), whereinc_(x,MAX)<=nc_(x)<=c_(x,MIN), and nc_(x) is at least as close to saidone of said c_(x,MIN) and said c_(x,MAX) determined in said determiningstep as to the other of said c_(x,MIN) and said c_(x,MAX); (f)outputting a result obtained using at least one nc_(x), wherein saidresult is indicative of one of: (i) a purity, identity, amount, orstructure of the substance, wherein the collection X is the outcome ofthe assay of the substance; (ii) data for one or more pixels, whereinthe pixels are elements x of the collection X; and (iii) data for one ormore audio segments, wherein the segments are elements x of thecollection X.